The Hölder continuous subsolution theorem for complex Hessian equations
[Le théorème des sous-solutions Hölder continues pour les équations hessiennes complexes]
Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 981-1007.

Soit Ω n un domaine borné fortement m-pseudoconvexe (1mn) et μ une mesure de Borel positive de masse finie sur Ω. Nous démontrons que l’équation hessienne complexe (dd c u) m β n-m =μ sur Ω admet une solution Hölder continue sur Ω ¯ pour une donnée au bord Hölder continue si (et seulement si) elle admet une sous-solution Hölder continue sur Ω ¯. L’étape principale dans la résolution du problème consiste à établir une nouvelle estimation capacitaire, qui montre que la mesure m-hessienne complexe d’une fonction m-sous-harmonique Hölder continue sur Ω ¯ avec valeur au bord nulle est dominée par la capacité m-hessienne par rapport à Ω avec un exposant explicite τ>1.

Let Ω n be a bounded strongly m-pseudoconvex domain (1mn) and μ a positive Borel measure with finite mass on Ω. We solve the Hölder continuous subsolution problem for the complex Hessian equation (dd c u) m β n-m =μ on Ω. Namely, we show that this equation admits a unique Hölder continuous solution on Ω ¯ with given Hölder continuous boundary values if it admits a Hölder continuous subsolution on Ω ¯. The main step in solving the problem is to establish a new capacity estimate showing that the m-Hessian measure of a Hölder continuous m-subharmonic function on Ω ¯ with zero boundary values is dominated by the m-Hessian capacity with respect to Ω with an (explicit) exponent τ>1.

Reçu le : 2020-04-17
Accepté le : 2020-06-29
Publié le : 2020-07-16
DOI : https://doi.org/10.5802/jep.133
Classification : 31C45,  32U15,  32U40,  32W20,  35J96
Mots clés: Équations de Monge-Ampère complexes, équations hessienne complexes, problème de Dirichlet, problèmes d’obstacle, sous-extension maximale, capacités hessiennes
@article{JEP_2020__7__981_0,
     author = {Amel Benali and Ahmed Zeriahi},
     title = {The H\"older continuous subsolution theorem for complex Hessian equations},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {981-1007},
     doi = {10.5802/jep.133},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__981_0/}
}
Amel Benali; Ahmed Zeriahi. The Hölder continuous subsolution theorem for complex Hessian equations. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 981-1007. doi : 10.5802/jep.133. https://jep.centre-mersenne.org/item/JEP_2020__7__981_0/

[AV10] S. Alesker & M. Verbitsky - “Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds”, Israel J. Math. 176 (2010), p. 109-138 | Article | Zbl 1193.53118

[BD12] R. J. Berman & J.-P. Demailly - “Regularity of plurisubharmonic upper envelopes in big cohomology classes”, in Perspectives in analysis, geometry, and topology, Progress in Math., vol. 296, Birkhäuser/Springer, New York, 2012, p. 39-66 | Article | Zbl 1258.32010

[Ber19] R. J. Berman - “From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit”, Math. Z. 291 (2019) no. 1-2, p. 365-394 | Article | Zbl 07051966

[Bre59] H. J. Bremermann - “On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries”, Trans. Amer. Math. Soc. 91 (1959), p. 246-276 | Article | Zbl 0091.07501

[BT76] E. Bedford & B. A. Taylor - “The Dirichlet problem for a complex Monge-Ampère equation”, Invent. Math. 37 (1976) no. 1, p. 1-44 | Article | Zbl 0315.31007

[BT82] E. Bedford & B. A. Taylor - “A new capacity for plurisubharmonic functions”, Acta Math. 149 (1982) no. 1-2, p. 1-40 | Article | Zbl 0547.32012

[Bło05] Z. Błocki - “Weak solutions to the complex Hessian equation”, Ann. Inst. Fourier (Grenoble) 55 (2005) no. 5, p. 1735-1756 | Article | Numdam | Zbl 1081.32023

[Ceg04] U. Cegrell - “The general definition of the complex Monge-Ampère operator”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 1, p. 159-179 | Article | Numdam | Zbl 1065.32020

[Cha16a] M. Charabati - Le problème de Dirichlet pour l’équation de Monge-Ampère complexe, Ph. D. Thesis, Université de Toulouse 3, 2016 | theses.fr:2016TOU30001

[Cha16b] M. Charabati - “Modulus of continuity of solutions to complex Hessian equations”, Internat. J. Math. 27 (2016) no. 1, article ID 1650003, 24 pages | Article | Zbl 1337.32046

[CIL92] M. G. Crandall, H. Ishii & P.-L. Lions - “User’s guide to viscosity solutions of second order partial differential equations”, Bull. Amer. Math. Soc. (N.S.) 27 (1992) no. 1, p. 1-67 | Article | Zbl 0755.35015

[CZ19] J. Chu & B. Zhou - “Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds”, Sci. China Math. 62 (2019) no. 2, p. 371-380 | Article | Zbl 1409.32029

[DDG + 14] J.-P. Demailly, S. Dinew, V. Guedj, H. H. Pham, S. Kołodziej & A. Zeriahi - “Hölder continuous solutions to Monge-Ampère equations”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 4, p. 619-647 | Article | Zbl 1296.32012

[DGZ16] S. Dinew, V. Guedj & A. Zeriahi - “Open problems in pluripotential theory”, Complex Var. Elliptic Equ. 61 (2016) no. 7, p. 902-930 | Article | Zbl 1345.32040

[DK14] S. Dinew & S. Kołodziej - “A priori estimates for complex Hessian equations”, Anal. PDE 7 (2014) no. 1, p. 227-244 | Article | Zbl 1297.32020

[EGZ09] P. Eyssidieux, V. Guedj & A. Zeriahi - “Singular Kähler-Einstein metrics”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 607-639 | Article | Zbl 1215.32017

[EGZ11] P. Eyssidieux, V. Guedj & A. Zeriahi - “Viscosity solutions to degenerate complex Monge-Ampère equations”, Comm. Pure Appl. Math. 64 (2011) no. 8, p. 1059-1094 | Article | Zbl 1227.32042

[GKZ08] V. Guedj, S. Kolodziej & A. Zeriahi - “Hölder continuous solutions to Monge-Ampère equations”, Bull. London Math. Soc. 40 (2008) no. 6, p. 1070-1080 | Article | Zbl 1157.32033

[GLZ19] V. Guedj, C. H. Lu & A. Zeriahi - “Plurisubharmonic envelopes and supersolutions”, J. Differential Geom. 113 (2019) no. 2, p. 273-313 | Article | Zbl 07122209

[GZ17] V. Guedj & A. Zeriahi - Degenerate complex Monge-Ampère equations, EMS Tracts in Math., vol. 26, European Mathematical Society, Zürich, 2017 | Article | Zbl 1373.32001

[KN20a] S. Kołodziej & N. C. Nguyen - “An inequality between complex hessian measures of Hölder continuous m-subharmonic functions and capacity”, in Geometric analysis, Progress in Math., vol. 333, Birkhäuser, Cham, 2020, p. 157-166 | Article | Zbl 07189286

[KN20b] S. Kołodziej & N. C. Nguyen - “A remark on the continuous subsolution problem for the complex Monge-Ampère equation”, Acta Math. Vietnam. 45 (2020) no. 1, p. 83-91 | Article | Zbl 07189286

[Koł96] S. Kołodziej - “Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator”, Ann. Polon. Math. 65 (1996) no. 1, p. 11-21 | Article | Zbl 0878.32014

[Koł05] S. Kołodziej - The complex Monge-Ampère equation and pluripotential theory, Mem. Amer. Math. Soc., vol. 178, no. 840, American Mathematical Society, Providence, RI, 2005 | Article | Zbl 1084.32027

[Li04] S.-Y. Li - “On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian”, Asian J. Math. 8 (2004) no. 1, p. 87-106 | Article | Zbl 1068.32024

[LPT20] C. H. Lu, T. T. Phung & T. D. Tô - “Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact hermitian manifolds”, 2020 | arXiv:2003.08417

[Lu12] C. H. Lu - Équations hessiennes complexes, Ph. D. Thesis, Université de Toulouse 3, 2012 | theses.fr:2012TOU30154

[Lu13] C. H. Lu - “Viscosity solutions to complex Hessian equations”, J. Funct. Anal. 264 (2013) no. 6, p. 1355-1379 | Article | Zbl 1266.32048

[Lu15] C. H. Lu - “A variational approach to complex Hessian equations in n ”, J. Math. Anal. Appl. 431 (2015) no. 1, p. 228-259 | Article | Zbl 1326.32056

[Ngu12] N. C. Nguyen - “Subsolution theorem for the complex Hessian equation”, Univ. Iagel. Acta Math. 50 (2012), p. 69-88 | Zbl 1295.32043

[Ngu14] N. C. Nguyen - “Hölder continuous solutions to complex Hessian equations”, Potential Anal. 41 (2014) no. 3, p. 887-902 | Article | Zbl 1302.32034

[Ngu18] N. C. Nguyen - “On the Hölder continuous subsolution problem for the complex Monge-Ampère equation”, Calc. Var. Partial Differential Equations 57 (2018) no. 1, article ID 8, 15 pages | Article | Zbl 1388.32029

[Ngu20] N. C. Nguyen - “On the Hölder continuous subsolution problem for the complex Monge-Ampère equation, II”, Anal. PDE 13 (2020) no. 2, p. 435-453 | Article | Zbl 07181506

[Pli14] S. Plis - “The smoothing of m-subharmonic functions”, 2014 | arXiv:1312.1906v2

[SA13] A. Sadullaev & B. Abdullaev - “Capacities and Hessians in the class of m-subharmonic functions”, Dokl. Akad. Nauk 448 (2013) no. 5, p. 515-517 | Article

[Sic81] J. Siciak - “Extremal plurisubharmonic functions in n ”, Ann. Polon. Math. 39 (1981), p. 175-211 | Article | Zbl 0477.32018

[SW08] J. Song & B. Weinkove - “On the convergence and singularities of the J-flow with applications to the Mabuchi energy”, Comm. Pure Appl. Math. 61 (2008) no. 2, p. 210-229 | Article | Zbl 1135.53047

[Tos18] V. Tosatti - “Regularity of envelopes in Kähler classes”, Math. Res. Lett. 25 (2018) no. 1, p. 281-289 | Article | Zbl 1397.32005

[Wal69] J. B. Walsh - “Continuity of envelopes of plurisubharmonic functions”, J. Math. Mech. 18 (1968/1969), p. 143-148 | Article | Zbl 0159.16002

[Zer20] A. Zeriahi - “Remarks on the modulus of continuity of subharmonic functions” (2020), Preprint available at http://www.math.univ-toulouse.fr/~zeriahi

[ÅCK + 09] P. Åhag, U. Cegrell, S. Kołodziej, H. H. Phạm & A. Zeriahi - “Partial pluricomplex energy and integrability exponents of plurisubharmonic functions”, Adv. Math. 222 (2009) no. 6, p. 2036-2058 | Article | Zbl 1180.31011